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Non-Homogeneous Effects on Spatiotemporal Chaos in Coupled Logistic Maps

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Non-Homogeneous Effects on Spatiotemporal Chaos in Coupled Logistic Maps Proceedings of COBEM 2009 20th International Congress of Mechanical Engineering Copyright © 2009 by ABCM November 15-20, 2009, Gramado, RS, Brazil NON-HOMOGENEOUS EFFECTS ON SPATIOTEMPORAL CHAOS IN COUPLED LOGISTIC MAPS Andre Varella Guedes, [email protected] Marcelo Amorim Savi, [email protected] Universidade Federal do Rio de Janeiro COPPE – Department of Mechanical Engineering 21.941.972 – Rio de Janeiro – RJ, Brazil, P.O. Box 68.503 Abstract. Spatiotemporal characteristics of a dynamical system are one of the important aspects related to the complexity of natural systems. This research effort deals with the spatiotemporal dynamics of coupled logistic maps. The logistic map lattice is coupled from a power law. Non-homogeneous behaviors of the grid are of concern evaluating the spatial interaction of different kinds of behaviors. Basically, the grid is split in two parts where each one can present qualitative different responses when isolated. Under this condition, the global dynamics is investigated evaluating how both parts interact to each other. Periodic boundary conditions are analyzed assuming that the values of the maps are repeated for every N maps. The influence of initial conditions is also of concern. Results show situations involving periodic and chaotic patterns. Keywords: Chaos, logistic maps, Lyapunov exponent, entropy, complexity, pattern formation. 1. INTRODUCTION Complexity is a term that is being used to denote the main characteristics of complex systems. In general, complexity of natural systems has characteristics such as self-organization and adaptive abilities that leads to pattern formation. Order and chaos are both related to this complexity and the balance between them occurs in the edge of chaos where “life has enough stability to sustain itself and enough creativity to be named as life” (Waldrop, 1992). The edge of chaos defines a region of spontaneity that is proper to life. Spatiotemporal characteristic of a dynamical system is one of the important aspects related to the complexity of natural systems. Literature presents numerous investigations concerning coupled maps. The key motivation is the search for universal properties and behaviors that apply to all dynamical systems. Therefore, coupled maps could be understood as prototypes of high dimensional dynamical systems. In general, synchronization of spatiotemporal dynamics is most intensively studied and there is a lack concerning the systematic investigation of other aspects of the system dynamics (Chazottes & Fernandes, 2005). In this regard, it is important to evaluate other aspects related to spatiotemporal dynamics. This article deals with the spatiotemporal dynamics of coupled maps. These maps represent a mathematical idealization of physical systems that are discrete in time and space and have been used to describe the evolution and pattern formation in different systems as chemical reactions, turbulence, neural networks and population dynamics. Specifically, this work is focused on a system composed by coupled logistic maps where coupling is described by a power law. Therefore, each map has the influence of other maps from its neighborhood and boundary conditions are also important to define the coupling characteristics. Periodic boundary conditions where the values of the maps are repeated for every N maps is of concern. The comparison between responses is made by the observation of the dynamics and the values of the Kolmogorov-Sinai entropy density. The influence of initial conditions is also treated. Non-homogeneous behaviors of the grid are of concern evaluating the spatial interaction of different kinds of behaviors. Basically, the grid is split in two parts where each one can present qualitative different responses when isolated. Under this condition, the global dynamics is investigated evaluating how both parts interact to each other. Numerical simulations allow one to conclude what types of conditions present greater tendency to develop chaotic, periodic and synchronized responses. 2. COUPLED MAPS The logistic map is employed to describe different problems related to economic and social areas. The logistic map is a first order difference equation represented by: xn+1 = f (xn ;β ) = βxn 1( − xn ) . Recently, coupled logistic maps are being used in order to model the evolution and pattern formation in systems associated with chemical reactions, turbulence, neural networks and population dynamics (Holden & Zhang, 1992). This work used a grid of N logistic maps where coupling is described by a power law as follows (Viana et al. , 2005): ε N ' 1 x (i) = 1− ε f x (i) + f x(i+ j) + f x(i− j) (1) n+1 () ()n ∑ α []()()n n η(α) j=1 j Proceedings of COBEM 2009 20th International Congress of Mechanical Engineering Copyright © 2009 by ABCM November 15-20, 2009, Gramado, RS, Brazil where N’=( N −1)/2 and f (x) = βx 1( − x) ; ε is the coupling intensity ( 0 ≤ ε ≤1 ), α is the coupling coverage (α ≥ 0 ), and η is given by: N' η(α) = 2∑ j −α (2) j=1 Each map i depends of its neighbors and the boundary conditions define the maps when i > N e i < 1. Figure 1 represents the coupled maps grid, illustrating the boundaries. i=1 i=N Figure 1. Coupled map grid. A numerous of boundary conditions can be established in order to represent distinct physical situations. The periodic condition assumes an infinite space where the maps repeat for each N maps. Mathematically, this condition is represented by: (i) (i±N ) xn = xn (3) Lyapunov spectrum represents one of the most important geometrical invariant of a dynamical system. The knowledge of this spectrum allows one to evaluate other invariants as the Kolmogorov-Sinai entropy. The analysis of Lyapunov spectrum in coupled maps can use the same methodology employed for a single map. Hence, let us assume coupled maps expressed by: (i) )1( K (i) K (N ) xn+1 = f (xn , , xn , , xn ) (4) The Lyapunov exponents determination needs to consider the variation of each cell under some perturbation in (i) initial conditions, x0 . The Jacobian matrix is calculated to each iteration as follows (Shibata, 2001), x )1( x )1( x )1( ∂ n+1 ∂ n+1 L ∂ n+1 )1( )2( ( N ) ∂xn ∂xn ∂xn x )2( x )2( x )2( ∂ n+1 ∂ n+1 L ∂ n+1 )1( )2( ( N ) J n = ∂xn ∂xn ∂xn (5) M M O M x( N ) x( N ) x( N ) ∂ n+1 ∂ n+1 L ∂ n+1 )1( )2( ( N ) ∂xn ∂xn ∂xn Then, defining n Rn = ∏ J k (6) k=1 (i) (i) Lyapunov exponents, λ , are evaluated from the eigenvalue σ of Rn, as follows (Holden & Zhang, 1992): 1 λ(i) = lim ln σ (i) (7) n→∞ n By considering the specific situation of coupled logistic maps where the coupling is defined by power law as presented in Eq.(1), the Jacobian matrix is written by: ( j) [J n ]ij = Aij f ′(xn ) (8) where A is a coupling dependent matrix (Batista & Viana, 2001): Proceedings of COBEM 2009 20th International Congress of Mechanical Engineering Copyright © 2009 by ABCM November 15-20, 2009, Gramado, RS, Brazil 1− ε if, i = j −α Aij ()ε,α = ε i − j /η(α), if i − j ≤ N′ (9) −α ε ()N′ − i − j /η(α), if i − j > N′ At this point, it is possible to use the classical algorithm due to Wolf et al. (1985) that uses the Gram-Schmidt ortonormalization of vectors of the tangent space (Lu et al. , 2005): v1 v 2 − (v2 .e1)e1 e1 = e2 = (10) v1 v 2 − (v2 .e1)e1 This approach allows the evaluation of the principal directions of the ellipsoid centered at a fiducial trajectory. The (i) norms of the orthonormalized vectors at the denominator of N k are used to calculate the Lyapunov exponents. Therefore, after n iterations: 1 n λ(i) = ln N (i) (11) n n ∑ k k=1 The Kolmogorov-Sinai entropy density is an index that can be calculated from the positive Lyapunov exponents as follows: 1 N ,λ>0 h = λ (12) N ∑ i i=1 Hence, when the entropy density vanishes there is no positive Lyapunov exponent and, therefore, there is no chaos. On the other hand, positive values of the entropy are related to chaos. 3. DYNAMICAL ANALYSIS Numerical simulations are carried out by assuming a grid with N = 21 and non-homogeneous values of parameter β through the grid. In general, the grid is split in two parts and each one has different value of this parameters. Basically, we need to analyze the spatial interaction between two qualitative different behaviors. The left side is defined from i = 1 to i = 11 being related to parameter βL, while the right side is defined from i = 12 to i = 21 being related to parameter βR. Values of parameter β are chosen in order to consider different qualitative behaviors of the isolated map, that can be indentified from a bifurcation diagram presented in Figure 2: 0 (period-1, stationary); 3.2 (period-2), 3.55 (period-8); 3.6, 3.7, 3.8 (chaotic); 3.835 (period-3 – periodic-window); 3.9, 4 (chaotic). All simulations are conducted assuming that parameter ε is between 0 and 1, while the parameter α is between 0 and 3. Periodic boundary conditions are focused on and different kinds of initial conditions are imposed to the system. Figure 2. Logistic map bifurcation diagram. Proceedings of COBEM 2009 20th International Congress of Mechanical Engineering Copyright © 2009 by ABCM November 15-20, 2009, Gramado, RS, Brazil Initially, let us consider an interaction between two dramatic behaviors: stationary ( βL = 0) and chaotic ( βR = 4). By assuming coupling parameters α = 3 and ε = 0.165, the homogeneous grid with β = 4 presents a chaotic behavior with h = 0.277 over a period-2 dynamics. On the other hand, the dynamics with β = 0 is stationary.
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